Propositional Proof Complexity Research Articles

A combinatorial characterization of resolution width

We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum space of refuting a 3-CNF formula is always bounded from below by the minimum width of refuting it (minus 3). This solves a well-known open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the dense linear order principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the relationship between size and width cannot be made subpolynomial.

The Complexity of Propositional Proofs with the Substitution Rule

The Complexity of Propositional Proofs with the Substitution Rule

The proof complexity of linear algebra

We introduce three formal theories of increasing strength for linear algebra in order to study the complexity of the concepts needed to prove the basic theorems of the subject. We give what is apparently the first feasible proofs of the Cayley–Hamilton theorem and other properties of the determinant, and study the propositional proof complexity of matrix identities such as AB= I→ BA= I.

On sufficient conditions for unsatisfiability of random formulas

A descriptive complexity approach to random 3-SAT is initiated. We show that unsatisfiability of any significant fraction of random 3-CNF formulas cannot be certified by any property that is expressible in Datalog. Combined with the known relationship between the complexity of constraint satisfaction problems and expressibility in Datalog, our result implies that any constraint propagation algorithm working with small constraints will fail to certify unsatisfiability almost always. Our result is a consequence of designing a winning strategy for one of the players in the existential pebble game. The winning strategy makes use of certain extension axioms that we introduce and hold almost surely on a random 3-CNF formula. The second contribution of our work is the connection between finite model theory and propositional proof complexity. To make this connection explicit, we establish a tight relationship between the number of pebbles needed to win the game and the width of the Resolution refutations. As a consequence to our result and the known size--width relationship in Resolution, we obtain new proofs of the exponential lower bounds for Resolution refutations of random 3-CNF formulas and the Pigeonhole Principle.

DEHN FUNCTION AND LENGTH OF PROOFS

We link the Dehn function of finitely presented groups to the length-of-proofs function in propositional proof complexity.

Propositional proof complexity

Propositional proof complexity

Berkowitz's algorithm and clow sequences

A combinatorial interpretation of Berkowitz's algorithm is presented. Berkowitz's algorithm is the fastest known parallel algorithm for computing the characteristic polynomial of a matrix. The combinatorial interpretation is based on loop covers introduced by Valiant, and sequences, defined by Mahajan and Vinay. Clow sequences turn out to capture very succinctly the computations performed by Berkowitz's algorithm, which otherwise are quite difficult to analyze. The main contribution of this paper is a proof of correctness of Berkowitz's algorithm in terms of clow sequences. multiple of log 2 n. There are two other fast parallel algorithms for computing the coefficients of the characteristic polynomial of a matrix: Chistov's algorithm and Csanky's algorithm. Chistov's algorithm is more difficult to formalize, and Csanky's algorithm works only for fields of char 0; see (8, section 13.4) for details about these two algorithms. The author's original motivation for studying Berkowitz's algorithm was the proof complexity of linear algebra. Proof Complexity deals with the complexity of formal mathematical derivations, and it has applications in lower bounds and automated theorem proving. In particular, the author was interested in the complexity of deriva- tions of matrix identities such as AB = I → BA = I (right matrix inverses are left inverses). These identities have been proposed by Cook as candidates for separating the Frege and Extended Frege proof systems. Proving (or disproving) this separation is one of the outstanding problems in propositional proof complexity; see (3) for a comprehensive exposition of this area of research. Thus, we were interested in an algorithm that could compute inverses of matrices, of the lowest complexity possible. Berkowitz's algorithm is ideal for our purposes for several reasons: • as was mentioned above, it has the lowest known complexity for computing

Generating hard tautologies using predicate logic and the symmetric group

We introduce methods to generate uniform families of hard propositional tautologies. The tautologies are essentially generated from a single propositional formula by a natural action of the symmetric group Sn. The basic idea is that any Second Order Existential sentence Ψ can be systematically translated into a conjunction φ of a finite collection of clauses such that the models of size n of an appropriate Skolemization Ψ are in one-to-one correspondence with the satisfying assignments to φn: the Sn-closure of φ, under a natural action of the symmetric group Sn. Each φn is a CNF and thus has depth at most 2. The size of the φn’s is bounded by a polynomial in n. Under the assumption NEXPTIME 6= coNEXPTIME, for any such sequence φn for which the spectrum S := {n : φn satisfiable} is NEXPTIME-complete, the tautologies ¬φn 6∈S do not have polynomial length proofs in any propositional proof system. Our translation method shows that most sequences of tautologies being studied in propositional proof complexity can be systematically generated from Second Order Existential sentences The International PhD Research School at BRICS, Aarhus, Denmark; Email: smriis@daimi.aau.dk Part of this work was done while visiting the Fields Institute, Toronto, Canada CISE Department, University of Florida, Gainesville, FL 32611-6120; Email: sitharam@cise.ufl.edu Supported in part by NSF Grant CCR 94-09809.

Relative efficiency of propositional proof systems: resolution vs. cut-free LK

Resolution and cut-free LK are the most popular propositional systems used for logical automated reasoning. The question whether or not resolution and cut-free LK have the same efficiency on the system of CNF formulas has been asked and studied since 1960 (Reckhow, Ph.D. Thesis, University of Toronto, 1976; A. Urquhart, the complexity of propositional proofs, Bull. of Symbolic Logic 1 (1995) 425–467). It was shown in Cook and Reckhow, J. Symbolic Logic 44 (1979) 36–50 that tree resolution has super-polynomial speed-up over (tree) cut-free LK. Naturally, the current issue is whether or not resolution and cut-free LK expressed as directed acyclic graphs (DAG) have the same efficiency. In this paper, we introduce a new algorithm to eliminate atomic cuts and show that cut-free LK (DAG) polynomially simulates resolution when the input formula is expressed as a k-CNF formula. As a corollary, we show that regular resolution does not polynomially simulate cut-free LK (DAG). We also show that cut-free LK (DAG) polynomially simulates regular resolution.

A Complexity Gap for Tree-Resolution

<p>It is shown that any sequence psi_n of tautologies which expresses the<br />validity of a fixed combinatorial principle either is "easy" i.e. has polynomial<br />size tree-resolution proofs or is "difficult" i.e requires exponential<br />size tree-resolution proofs. It is shown that the class of tautologies which<br />are hard (for tree-resolution) is identical to the class of tautologies which<br />are based on combinatorial principles which are violated for infinite sets.<br />Actually it is shown that the gap-phenomena is valid for tautologies based<br />on infinite mathematical theories (i.e. not just based on a single proposition).<br />We clarify the link between translating combinatorial principles (or<br />more general statements from predicate logic) and the recent idea of using<br /> the symmetrical group to generate problems of propositional logic.<br />Finally, we show that it is undecidable whether a sequence psi_n (of the<br />kind we consider) has polynomial size tree-resolution proofs or requires<br />exponential size tree-resolution proofs. Also we show that the degree of<br />the polynomial in the polynomial size (in case it exists) is non-recursive,<br />but semi-decidable.</p><p>Keywords: Logical aspects of Complexity, Propositional proof complexity,<br />Resolution proofs.</p><p> </p>

Generating Hard Tautologies Using Predicate Logic and the Symmetric Group

We introduce methods to generate uniform families of hard propositional tautologies. The tautologies are essentially generated from a single propositional formula by a natural action of the symmetric group Sn.<br />The basic idea is that any Second Order Existential sentence Psi can be systematically translated into a conjunction phi of a finite collection of clauses such that the models of size n of an appropriate Skolemization Psi~ are in one-to-one correspondence with the satisfying assignments to phi_n: the Sn-closure of phi, under a natural action of the symmetric group Sn. Each phi_n is a CNF and thus has depth at most 2. The size of the phi_n's is bounded by a polynomial in n. Under the assumption NEXPTIME |= co- NEXPTIME, for any such sequence phi_n for which the spectrum S := {n : phi_n satisfiable} is NEXPTIME-complete, the tautologies not phi_(n not in S) do not have polynomial length proofs in any propositional proof system.<br /> Our translation method shows that most sequences of tautologies being studied in propositional proof complexity can be systematically generated from Second Order Existential sentences and moreover, many natural mathematical statements can be converted into sequences of propositional tautologies in this manner.<br /> We also discuss algebraic proof complexity issues for such sequences of tautologies. To this end, we show that any Second Order Existential sentence Psi can be systematically translated into a finite collection of polynomial equations Q = 0 such that the models of size n of an appropriate skolemization Psi~ are in one-to-one correspondence with the solutions to Qn = 0: the Sn-closure of Q = 0, under a natural action of the symmetric group Sn. The degree of Qn is the same as that of Q, and hence is independent of n, and the number of variables is no more than a polynomial in n. Furthermore, using results in [19] and [20], we briefly describe how, for the corresponding sequences of tautologies phi_n, the rich structure of the Sn closed, uniformly generated, algebraic systems Qn has profound consequences on the algebraic proof complexity of phi_n.

The Complexity of Propositional Proofs

§1. Introduction. The classical propositional calculus has an undeserved reputation among logicians as being essentially trivial. I hope to convince the reader that it presents some of the most challenging and intriguing problems in modern logic. Although the problem of the complexity of propositional proofs is very natural, it has been investigated systematically only since the late 1960s. Interest in the problem arose from two fields connected with computers, automated theorem proving and computational complexity theory. The earliest paper in the subject is a ground-breaking article by Tseitin [62], the published version of a talk given in 1966 at a Leningrad seminar. In the three decades since that talk, substantial progress has been made in determining the relative complexity of proof systems, and in proving strong lower bounds for some restricted proof systems. However, major problems remain to challenge researchers. The present paper provides a survey of the field, and of some of the techniques that have proved successful in deriving lower bounds on the complexity of proofs. A major area only touched upon here is the proof theory of bounded arithmetic and its relation to the complexity of propositional proofs. The reader is referred to the book by Buss [10] for background in bounded arithmetic. The forthcoming book by Krajíček [40] also gives a good introduction to bounded arithmetic, as well as covering most of the basic results in complexity of propositional proofs.